Question

simplify the expression using the properties of exponents.
\\((2a^{-2}c^4)^0(3a^4c^3)^2\\)
show your work here
hint: to add an exponent \\((x^y)^z\\), type \exponent\ or press \^\
enter your answer

Explanation:

Step1: Simplify \((2a^{-2}c^4)^0\)

Any non - zero number to the power of 0 is 1. Since \(2a^{-2}c^4\) is non - zero (assuming \(a
eq0\) and \(c
eq0\)), \((2a^{-2}c^4)^0 = 1\)

Step2: Simplify \((3a^{4}c^{3})^{2}\)

Using the power of a product rule \((xy)^n=x^n y^n\) and the power of a power rule \((x^m)^n=x^{mn}\), we have:
For the coefficient: \(3^{2}=9\)
For the \(a\) term: \((a^{4})^{2}=a^{4\times2}=a^{8}\)
For the \(c\) term: \((c^{3})^{2}=c^{3\times2}=c^{6}\)
So \((3a^{4}c^{3})^{2}=9a^{8}c^{6}\)

Step3: Multiply the two simplified expressions

We have \((2a^{-2}c^4)^0\times(3a^{4}c^{3})^{2}=1\times9a^{8}c^{6}=9a^{8}c^{6}\)

Answer:

\(9a^{8}c^{6}\)